Realizability theory for continuous linear systems, Volume 97 | Date: 27 April 2011, 11:50
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Realizability theory for continuous linear systems, Volume 97 (Mathematics in Science and Engineering)
By Zemanian
* Publisher: Academic Press
* Number Of Pages: 231
* Publication Date: 1972-02-11
* ISBN-10 / ASIN: 0127795502
* ISBN-13 / EAN: 9780127795508
Product Description:
Concise exposition of realizability theory as applied to continous linear systems, specifically to the operators generated by physical systems as mappings of stimuli into responses. Many problems included.
Preface:
“Realizability theory” is a part of mathematical systems theory and is
concerned with the following ideas. Any physical system defines a relation
between the stimuli imposed on the system and the corresponding responses.
Moreover, any such system is always causal and may possess other properties
such as time-invariance and passivity. Two questions: How are the physical
properties of the system reflected in various analytic descriptions of the
relation? Conversely, given an analytic description of a relation, does there
exist a corresponding physical system possessing certain specified properties ?
If the latter is true, the analytic description is said to be realizable. Considerations
of this sort arise in a number of physical sciences. For example, see
McMillan (1952), Newcomb (1966), or Wohlers (1969) for electrical networks,
Toll (1956) or Wu (1954) for scattering phenomena, and Gross (1953), Love
(1956), or Meixner (1954) for viscoelasticity.
This book is an exposition of realizability theory as applied to the operators
generated by physical systems as mappings of stimuli into responses. This
constitutes the so-called “ black box ” approach since we do not concern ourselves
with the internal structure of the system at hand. Physical characteristics
such as linearity, causality, time-invariance, and passivity are defined as
mathematical restrictions on a given operator. Then, the two questions are
answered by obtaining a description of the operator in the form of a kernel
or convolution representation and establishing a variety of necessary and
sufficient conditions for that representation to possess the indicated properties.
Thus, the present work is an abstraction of classical realizability theory
in the following way. A given representation is realized not by a physical system
but rather by an operator possessing mathematically defined properties, such
as causality and passivity, which have physical significance. We may state
this in another way. Our primary concern is the study of physical properties
and their mathematical characterizations and not the design of particular
systems.
Two properties we shall always impose on any operator under consideration
are linearity and continuity. They are quite commonly (but by no means
always) possessed by physical systems. Of course, continuity only has a
meaning with respect to the topologies of the domain and range spaces of the
operator. We can in general take into account a wider class of continuous
linear operators by choosing a smaller domain space with a stronger topology
and a larger range space with a weaker topology. With this as our motivation,
we choose the basic testing-function space of distribution theory as the domain
for our operators and the space of distributions as the range space. The
imposition of other properties upon the operator will in general allow us to
extend the operator onto wider domains in a continuous fashion. For example,
time-invariance implies that the operator has a convolution representation
and can therefore be extended onto the space of all distributions with compact
supports. This distributional setting also provides the following facility. It
allows us to obtain certain results, such as Schwartz’s kernel theorem, which
simply do not hold under any formulation that permits the use of only
ordinary functions. Thus, distribution theory provides a natural language for
the realizability theory of continuous linear systems.
Still another facet of this book should be mentioned. Almost all the realizability
theories for electrical systems deal with signals that take their instantaneous
values in n-dimensional Euclidean space. However, there are
many systems whose signals have instantaneous values in a Hilbert or Banach
space. Section 4.2 gives an example of this. For this reason, we assume that the
domain and range spaces for the operator at hand consist of Banach-spacevalued
distributions. Many of the results of earlier realizability theories readily
carry over to this more general setting, other results go over but with difficulty,
and some do not generalize at all. Moreover, the theory of Banach-spacevalued
distributions is somewhat more complicated than that of scalar
distributions; Chapter 3 presents an exposition of it. Still other analytical
tools we shall need as a consequence of our use of Banach-space-valued
distributions are the elementary calculus of functions taking their values in
locally convex spaces, which is given in Chapter 1, and Hackenbroch’s
theory for the integration of Banach-space-valued functions with respect
to operator-valued measures, a subject we discuss in Chapter 2.
The systems theory in this book occurs in Chapters4,5,7, and 8. Chapter 4
is a development of Schwartz’s kernel theorem in the present context and ends
with a kernel representation for our continuous linear operators. Causality
appears as a support condition on the kernel. How time-invariance converts a
kernel operator into a convolution operator is indicated in Chapter 5. We
digress in Chapter 6 to develop those properties of the Laplace transformation
that will be needed in our subsequent frequency-domain discussions.
Passivity is a very strong assumption; it is from this that we get the richest
realizability theory. Chapter 7 imposes a passivity condition that is appropriate
for scattering phenomena, whereas a passivity condition that is suitable
for an admittance formulism is exploited in Chapter 8.
It is assumed that the reader is familiar with the material found in the
customary undergraduate courses on advanced calculus, Lebesgue integration,
and functions of a complex variable. Furthermore, a variety of
standard results concerning topological linear spaces and the Bochner integral
will be used. In order to make this book accessible to readers who
may be unfamiliar with either of these topics, a survey of them is given in
the appendixes. Although no proofs are presented, enough definitions and
discussions are given to make what is presented there understandable, it is
hoped, to someone with no knowledge of either subject. Almost every result
concerning the aforementioned two topics that is used in this book can be
found in the appendixes, and a reference to the particular appendix where it
occurs is usually given. For the few remaining results of this nature that are
employed, we provide references to the literature.
The problems usually ask the reader either to supply the proofs of certain
assertions that were made but not proved in the text or to extend the theory in
various ways. On occasion, we employ a result that was stated only in a
previous problem. For this reason, it is advisable for the reader to pay some
attention to the problems.
All theorems, corollaries, lemmas, examples, and figures are triplenumbered
; the first two numbers coincide with the corresponding section
numbers. On the other hand, equations are single-numbered starting with
(1) in each section.
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